Wave Properties: Speed, Amplitude, Frequency, and Period

13.2 Wave Properties: Speed, Amplitude, Frequency, and Period

Section Learning Objectives

By the end of this section, you will be able to do the following:

Define amplitude, frequency, period, wavelength, and velocity of a wave
Relate wave frequency, period, wavelength, and velocity
Solve problems involving wave properties

Teacher Support

The learning objectives in this section will help your students master the following standards:

(7) Science concepts. The student knows the characteristics and behavior of waves. The student is expected to:
- (B) investigate and analyze the characteristics of waves, including velocity, frequency, amplitude, and wavelength, and calculate using the relationship between wave speed, frequency, and wavelength;
- (D) investigate the behaviors of waves, including reflection, refraction, diffraction, interference, resonance, and the Doppler effect.

Section Key Terms

wavelength

wave velocity

Teacher Support

[BL][OL][AL] Review amplitude, period, and frequency for simple harmonic motion.

Wave Variables

In the chapter on motion in two dimensions, we defined the following variables to describe harmonic motion:

Amplitude—maximum displacement from the equilibrium position of an object oscillating around such equilibrium position
Frequency—number of events per unit of time
Period—time it takes to complete one oscillation

For waves, these variables have the same basic meaning. However, it is helpful to word the definitions in a more specific way that applies directly to waves:

Amplitude—distance between the resting position and the maximum displacement of the wave
Frequency—number of waves passing by a specific point per second
Period—time it takes for one wave cycle to complete

In addition to amplitude, frequency, and period, their wavelength and wave velocity also characterize waves. The wavelength $λ$ is the distance between adjacent identical parts of a wave, parallel to the direction of propagation. The wave velocity $v_{w}$ is the speed at which the disturbance moves.

Tips For Success

Wave velocity is sometimes also called the propagation velocity or propagation speed because the disturbance propagates from one location to another.

Consider the periodic water wave in Figure 13.7. Its wavelength is the distance from crest to crest or from trough to trough. The wavelength can also be thought of as the distance a wave has traveled after one complete cycle—or one period. The time for one complete up-and-down motion is the simple water wave’s period T. In the figure, the wave itself moves to the right with a wave velocity v_w. Its amplitude X is the distance between the resting position and the maximum displacement—either the crest or the trough—of the wave. It is important to note that this movement of the wave is actually the disturbance moving to the right, not the water itself; otherwise, the bird would move to the right. Instead, the seagull bobs up and down in place as waves pass underneath, traveling a total distance of 2X in one cycle. However, as mentioned in the text feature on surfing, actual ocean waves are more complex than this simplified example.

A seagull bobs up and down on a sinusoidal-shaped periodic ocean wave with a given wave velocity of νw. The wavelength λ is shown as the distance from one crest to the next crest. The amplitude X is shown as the distance between the resting position and the crest. The total distance traveled by the seagull in one cycle or period is shown as 2X.

Figure 13.7 The wave has a wavelength λ, which is the distance between adjacent identical parts of the wave. The up-and-down disturbance of the surface propagates parallel to the surface at a speed v_w.

Watch Physics

Amplitude, Period, Frequency, and Wavelength of Periodic Waves

This video is a continuation of the video “Introduction to Waves” from the "Types of Waves" section. It discusses the properties of a periodic wave: amplitude, period, frequency, wavelength, and wave velocity.

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Tips For Success

The crest of a wave is sometimes also called the peak.

Watch Physics: Amplitude, Period, Frequency and Wavelength of Periodic Waves. This video introduces several concepts of sound; amplitude, period, frequency, and wavelength of periodic waves.

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In this animation, watch how a string vibrates in slow motion by choosing the Slow Motion setting. Select the No End and Manual options, and wiggle the end of the string to make waves yourself. Then switch to the Oscillate setting to generate waves automatically. Adjust the frequency and the amplitude of the oscillations to see what happens. Then experiment with adjusting the damping and the tension.

Grasp Check

Which of the settings—amplitude, frequency, damping, or tension—changes the amplitude of the wave as it propagates? What does it do to the amplitude?

Frequency; it decreases the amplitude of the wave as it propagates.
Frequency; it increases the amplitude of the wave as it propagates.
Damping; it decreases the amplitude of the wave as it propagates.
Damping; it increases the amplitude of the wave as it propagates.

Solving Wave Problems

Worked Example

Calculate the Velocity of Wave Propagation: Gull in the Ocean

Calculate the wave velocity of the ocean wave in the previous figure if the distance between wave crests is 10.0 m and the time for a seagull to bob up and down is 5.00 s.

Strategy

The values for the wavelength $(λ = 10.0 m)$ and the period $(T = 5.00 s)$ are given and we are asked to find $v_{w}$ Therefore, we can use $v_{w} = \frac{λ}{T}$ to find the wave velocity.

Solution

Enter the known values into $v_{w} = \frac{λ}{T}$

v_{w} = \frac{10 .0 m}{5 .00 s} = 2. 00 m/s .

13.5

Discussion

This slow speed seems reasonable for an ocean wave. Note that in the figure, the wave moves to the right at this speed, which is different from the varying speed at which the seagull bobs up and down.

Worked Example

Calculate the Period and the Wave Velocity of a Toy Spring

The woman in Figure 13.3 creates two waves every second by shaking the toy spring up and down. (a)What is the period of each wave? (b) If each wave travels 0.9 meters after one complete wave cycle, what is the velocity of wave propagation?

Strategy FOR (A)

To find the period, we solve for $T = \frac{1}{f}$ , given the value of the frequency $(f = 2 s^{- 1}).$

Solution for (a)

Enter the known value into $T = \frac{1}{f}$

T = \frac{1}{2 s^{−1}} = 0. 5 s .

13.6

Strategy FOR (B)

Since one definition of wavelength is the distance a wave has traveled after one complete cycle—or one period—the values for the wavelength $(λ = 0.9 m)$ as well as the frequency are given. Therefore, we can use $v_{w} = f λ$ to find the wave velocity.

Solution for (b)

Enter the known values into $v_{w} = f λ$

$v_{w} = f λ = (2 s^{−1})(0 .9 m) = 1 .8 m/s .$

Discussion

We could have also used the equation $v_{w} = \frac{λ}{T}$ to solve for the wave velocity since we already know the value of the period $(T = 0.5 s)$ from our calculation in part (a), and we would come up with the same answer.

Practice Problems

The frequency of a wave is 10 Hz. What is its period?

The period of the wave is 100 s.
The period of the wave is 10 s.
The period of the wave is 0.01 s.
The period of the wave is 0.1 s.

What is the velocity of a wave whose wavelength is 2 m and whose frequency is 5 Hz?

20 m/s
2.5 m/s
0.4 m/s
10 m/s

Check Your Understanding

Teacher Support

Use these questions to assess students’ achievement of the section’s Learning Objectives. If students are struggling with a specific objective, these questions will help identify such objective and direct them to the relevant content.

What is the amplitude of a wave?

A quarter of the total height of the wave
Half of the total height of the wave
Two times the total height of the wave
Four times the total height of the wave

10 .

What is meant by the wavelength of a wave?

The wavelength is the distance between adjacent identical parts of a wave, parallel to the direction of propagation.
The wavelength is the distance between adjacent identical parts of a wave, perpendicular to the direction of propagation.
The wavelength is the distance between a crest and the adjacent trough of a wave, parallel to the direction of propagation.
The wavelength is the distance between a crest and the adjacent trough of a wave, perpendicular to the direction of propagation.

11 .

How can you mathematically express wave frequency in terms of wave period?

$f = \frac{1}{T}$
$f = \left(\frac{1}{T}\right)^2$
$\text{f} = \text{T}$
$f = \left(T\right)^2$

12.

When is the wavelength directly proportional to the period of a wave?

When the velocity of the wave is halved
When the velocity of the wave is constant
When the velocity of the wave is doubled
When the velocity of the wave is tripled